Finding the power series of a rational functionIn many combinatorial enumeration problems it is possible...

Rebecca Villa

Rebecca Villa

Answered

2022-07-09

Finding the power series of a rational function
In many combinatorial enumeration problems it is possible to find a rational generating function (i.e. the quotient of two polynomials) for the sequence in question. The question is - given the generating function, how can we find (algorithmically) the values of the sequence, i.e. the coefficients of the corresponding power series?
I know that for a rational generating function, the sequence satisfies a recurrence relation given by the coefficients of the polynomial in the denominator, so it's really just the question of finding the finite initial values.

Answer & Explanation

Wade Atkinson

Wade Atkinson

Expert

2022-07-10Added 12 answers

If
f ( x ) = P ( x ) Q ( x )
where P , Q are polynomials, then we have that
f ( x ) Q ( x ) = P ( x )
Now use Leibniz's formula for differentiating a product
r = 0 n ( n r ) f r ( x ) Q n r ( x ) = P n ( x )
which you can evaluate at 0 for n = 1 , 2 , … and you can obtain ( n + 1 ) t h derivative of f at 0 from the first n derivatives using the above formula.
The coefficient you need would be f n ( 0 ) n ! .

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